Help me how to find the base of the intersection of two subspaces

Hello, Can someone help me how to find the base of the intersection of two vector subspaces in matlab? please

Thanks!

Muchas gracias!

Answer 1

[Matt J]

If the vectors form the columns of a matrix A and the vectors form the columns of a matrix B, then the basis of the intersecting space can be found as,

null( [null(A.').' ; null(B.').'] )

[Maria Christine Honecker]

Hello, as I would like to use this further, I would be interested to know if there is any proof or source for this?

I am grateful for any answer.

[Matt J]

@Maria Christine Honecker You will find in many/most linear algebra texts that the row space and the null space of a matrix are orthogonal complements. Therefore, if a linear space is described by matrix equations E*x=0 then null(E) gives a basis for the subspace. Conversly, if B is a basis for the subpace, then E=null(B.').' is its equation coefficient matrix.

Therefore, the equation matrix for subspace L is null(A.').', the equation matrix for subspace M is null(B.').', and the equation matrix for their intersection is the concatenated system [null(A.').' ; null(B.').']. A basis for the concatenated system is null([null(A.').' ; null(B.').']).